Doublespeak here ought to be doublethink. The person who first defined it must have had doublethink confused with newspeak and thus combined the two. The definition given is for doublethink.

Whatever was true now was true from everlasting to everlasting. It was quie simple All that was needed was an unending series of victories over your own memmory. "Reality control," they called it; in Newspeak, "doublethink."

From George Orwell's 1984. The guy was 20 years out, but other than that fairly accurate... It's the ability to hold two separated and completely contradictory statements in your head and believe that both are true

Dubya: "We are bringing democracy to Iraq" (while bringing a dictatorship to the US via the "Patroit" act, Florida elections etc etc etc).

The Urban Dictionary T-Shirt

Words combined into a sentence, paragraph, speech, etc. SAY one thing, yet MEAN something different. As the American Native Peoples called the white man's words - He speaks with forked tongue

The council has agreed to the demands of all those detained without charge; they will be allowed weekly phone calls with family members.
WHAT IS NOT SAID:::The calls will be monitored, and the charges will be so expensive that the common man/woman's family cannot pay for the call.

The Urban Dictionary T-Shirt

When you are messaging someone back and forth and while you are writing a response their last message they send another message on a different topic. You realize the second message requires a completely seperate message to answer so you quickly finish you initial response and send it. Then as you start typing the second message the other person reads the message you already sent then proceeds to write a message back, ignoring the second topic of conversation as if he or she were talking to someone else. You respond in the same fashion and the two of you have two seperate messaging conversations at the same time until some point one topic is dropped or the messaging ends altogether.

I was messaging Bill earlier and it was total doublespeak
Example
You: Want to see a movie later?
Me: Sure, what film?
Me again: Did you see how hot Kyla looked in those yoga pants today?
You: How about 'Spaceballs?'
You: Yeah man, she has some hot curves!
Me: Meet you downtown around eight?
Me: She sure does, those pants are so tight i bet she uses paint thinner to take them off